In the current study skins made from 0.8 mm thick plates of AA1050-H14 aluminum were used. AA1050A-H14 is a 1000-series aluminum alloy considered commercially pure and is formulated for primary forming into wrought products [52], produced by Hindalco Industries Ltd. It is usually formed by extrusion or rolling. The alloy-series can be strengthened by cold working, but not by heat treatment [53]. It is furnished in the H14 temper to develop a particular degree of intergranular and exfoliation corrosion. The nominal yield stress and ultimate tensile strengths were given by the manufacturer to be 110 M P a and 116 M P a respectively. The chemical composition of the alloy is shown in Table 4.1. Table 4.2 presents the physical material constants for aluminum used in the material models.
Table 4.1: Chemical composition of AA1050A-H14 [18].
Si [%] Fe [%] Cu [%] Mn [%] Mg [%] Zn [%] Ti [%] Al [%]
0.030 0.360 0.001 0.002 0.000 0.003 0.010 Rest
Table 4.2: Physical material constants for AA1050-H14 [1].
E v ρ α Cp χ Tr Tm
[GP a] [−] [kg/m3] [K−1] [J/kgK] [−] [K] [K] 70 0.33 2705 1.2∗10−5 452 0.9 293 893
To establish the material properties of the AA1050-H14 aluminum alloy used in the shock tube experiments, nine uni-axial tensile tests of UT-200 flat dog bone components created from the aluminum skins were performed at room temperature. Three tests were carried out for the 0-degree, 45-degree and 90-degree direction of the material, relative to the rolling direction.
The nominal geometry of the specimens is shown in Figure 4.1. The apparatus used for the tensile tests was an Instron 5985 with a load cell of250 kN. The tests were performed by fixing one end of the specimens while pulling the other. Both sides of the dog bone specimens were attached to the machine with a bolt through a hole in the specimen. The machine applied a displacement at a rate of 1.0 mm/min until the specimens reached fracture. Reaction force and displacement were measured by the machine, while a camera was used to capture the deformation of the test specimens. Displacement data retrieved from the Instron machine was disregarded due to error sources such as machine stiffness. The strain was obtained from image analysis using 2D-DIC with the software eCorr. Data for force and strain was post-processed to obtain the stress-strain relationship.
Figure 4.1: Dimensions of UT-200 flat dog bone test specimen.
4.1.1 Digital Image Correlation
DIC is an optical method that measures changes in an images series [54]. In this thesis, the DIC software eCorr was used to generate strain plots and local strain values in nodes. The specimens were painted with a random speckle pattern to improve the accuracy of the results. A camera was placed such that the optical axis was normal to the dog bone specimen surfaces. Using eCorr, a mesh of Q4 elements was placed on the test specimens in their initial configuration. By tracking the development in the pictures, the software calculated displacements and strains in the nodes of the mesh. Inaccuracies are usually due to grayscale noise in the recorded images.
The results can be improved by creating a vector over the uniform middle part of the specimen as a virtual extensometer to even out the deviations in the nodes.
4.1.2 Material Model
An elastic-plastic material model was used to predict the response of the aluminum alloy AA1050-H14 in this study. Von Mises yield criterion was considered in the material model due to its appliance to ductile materials and since it is widely used for uni-axial tension tests [55].
Von Mises assumes an isotropic, isochoric, isothermal and rate independent material where yielding is independent of the hydrostatic pressure. Several studies suggest an anisotropic behavior for extruded aluminum alloys [56]. Therefore three directions relative to the rolling of the material were tested and used for material model calibration. The work hardening was described using Voce hardening law. The tensile test data was processed and curve fitted to calibrate the hardening model for the three different directions. Cockcroft-Latham [37] fracture criterion was added to the material model and calibrated with trial and error numerically by comparison of the material model and the experimental results.
4.1.3 Elastic Properties
The force data obtained during the tensile tests were used to calculate engineering stress, σe. Engineering stress is a measurement of stress based on force and the initial cross-section area of the specimen as seen in Eq. 4.1, which means it disregards the area reduction during the elastic, and more significantly, during the plastic deformation. Calculated engineering stress for all material tests is shown in Figure 4.2. Notice that the third test for the 0-degree direction was disregarded due to an error in the experimental setup for that particular test. Spikes in the stress-strain curves can be observed. These were caused by yield in the specimen at the interaction between the bolt and the holed part of the specimen. However, the spikes are insignificant to the shape of the curves and will, for further calibration procedures, be disregarded. As seen in Figure 4.2, the variations within the material test for each direction are small. The second test for each direction was picked as a representative presentation of the behavior in each direction. These three tests will be used for further illustration of the calibration procedure described below.
True stress and true strain were calculated to include the deformation of the specimen and describe the response accurately. In Figure 4.3 it can be observed that the engineering stress is similar to the true stress in the elastic region but significantly lower in the plastic zone. This effect is due to the elevated changes to the specimens area in the plastic zone compared to the elastic zone. These measurements were derived assuming that the volume of the specimen was conserved, an isochoric material, as seen in Eq. 4.2 and that plastic deformation zone is far greater than elastic deformation zone. Eq. 4.3 and Eq. 4.4 were used to calculate true stress, σt, and true strain, εt, respectively, until necking. Necking is initiated at maximum force recorded.
σe = F
A0 (4.1)
A= A0L0
L (4.2)
σt =σe(1 +εe) (4.3)
εt= ln (1 +εe) (4.4)
Young’s Modulus, E, yield stress, σ0, and Poisson’s Ratio,ν, was not calibrated in this study, instead generally accepted values of 70 GP a, 80 M P a and 0.33, respectively, was used in further calculations [2, 56].
0.00 0.02 0.04 0.06 0.08 0.10
0.00 0.02 0.04 0.06 0.08 0.10
Strain [-]
0.00 0.02 0.04 0.06 0.08 0.10
Strain [-]
Figure 4.2: Engineering stress plotted with engineering strain for all tension tests for AA1050-H14 in the 0-degree, 45-degree and 90-degree direction, relative to the rolling direction of the material.
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Strain [-]
0.000 0.005 0.010 0.015 0.020
Strain [-]
Figure 4.3: Engineering stress, necking and yield stress plotted with engineering strain (left) and true stress plotted with true strain (right) for selected tension tests for AA1050-H14.
4.1.4 Plastic Properties
During blast loading, the aluminum skins may undergo large plastic deformation. Therefore, it is instrumental to have information about the material’s plastic behavior. The performed tensile tests only provide information pre-necking which occurs at maximum force, at approximately 0.01 to0.02elongation for all three directions. At necking, a biaxial stress state is introduced which distorts the results for uniaxial behavior. Therefore, the post-necking behavior was disregarded. The plastic strains, εpl, were obtained by subtracting elastic strains from the true strains as seen in Eq. 4.5, and was the basis for calibrating the hardening model.
εpl =ε−εel (4.5)
The true stress-plastic strain curve was approximated using a nonlinear least squares solver.
Voce hardening law was used to describe the work hardening of the material, where the least square solver provides the hardening parameters to the optimized curve fit. Due to the large deformations during the shock tube experiments, significantly larger strains than necking are expected. The stress post-necking is unknown but can be approximated by extrapolating the hardening laws as seen in Figure 4.4 (right). The Voce law is based on a series of exponential functions where Qi and Ci are optimized for each part of the series, shown in Eq. 4.6. The curve fits of the hardening law for all three directions are shown in Figure 4.4 (left). Two exponential functions were sufficient to provide an accurate fitting of the curves. Extrapolated results are seen in Figure 4.4 (right) while Table 4.3 shows the optimized hardening parameters for Voce hardening law.
σ =σ0+
n
X
i=1
Qi(1−e−Cip) (4.6)
0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 Strain [-]
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Strain [-]
Figure 4.4: Curve fits for Voce hardening law for all three directions (left) and extrapolated curve fits for strains post necking (right).
Table 4.3: Curve fitted Voce hardening law material model parameters for 0-degrees, 45-degrees and 90-degrees relative to the rolling direction of aluminum alloy AA1050-H14.
Direction Q1 C1 Q2 C2
0-degree 18.65 1884.49 10.36 185.73 45-degree 21.66 1702.63 11.66 251.62 90-degree 17.79 2427.84 15.16 339.68
4.1.5 Numerical Validation
A numerical model of the test specimen was created to validate the calibrated material model, as seen in Figure 4.5. The same constitutive model, element formulation and hourglass control as described in Section 3.2 was applied to the numerical model of the dog bone specimen. The geometry of the test specimen can be seen in Figure 4.1. The holed parts and the gauge section of the specimen are meshed with an element size of 3.0 mm and 0.8 mm, respectively. Fixed boundary conditions were applied to the left hole, while a velocity was applied to the right hole equivalent to a translation of 5mm during 151 seconds.
Figure 4.5: Numerical model of test specimen UT-200 flat dog bone.
The material parameters applied to the model described above can be seen in Table 4.4. Note that the Johnson-Cook hardening parameter, c, is equal zero to obtain the quasi-static condition of the material tests using a time-scaled numerical model. The Cockcroft-Latham fracture parameters were obtained by trial and error for the three material directions. Note that necking is a mesh sensitive phenomenon and that the element exposed to the largest plastic work, the critical element, is always located inside the neck [15]. Therefore, the fracture parameter is highly mesh dependent and is calibrated for the element size of the tensile test. Results on mesh refinement on aluminum skins without foam core presented in Section 3.3 shows that mesh size is a dominant factor when modeling fracture, but only affects the midpoint displacement to a small degree. Since the main scope of this study is to investigate the structural response of sandwich panels exposed to blast loading and not fracture, the fracture criterion calibrated for an element size of 0.8mm is assumed sufficient.
Table 4.4: Material parameters for the constitutive relation for AA1050-H14.
Direction σ0 Q1 C1 Q2 C2 c m p˙0 Wc
[M P a] [M P a] [−] [M P a] [−] [−] [−] [s−1] [M P a] 0-degree 80.0 18.65 1884.49 10.36 185.73 0.0 1.0 5∗10−4 60.0 45-degree 80.0 21.66 1702.63 11.66 251.62 0.0 1.0 5∗10−4 45.0 90-degree 80.0 17.79 2427.84 15.16 339.68 0.0 1.0 5∗10−4 25.0
The numerical work based on the obtained material parameters compared with the experimen-tal results is presented in Figure 4.6. An agreement is observed pre-necking, while post-necking the numerical results display a more sudden and complete fracture characteristic than the ex-perimental results. The model is assumed sufficient to study the structural response of sandwich panels exposed to blast loading.
0.00 0.01 0.02 0.03 0.04 0.05 0.06 Strain [-]
0 20 40 60 80 100
Stress [MPa]
0-degree 45-degree 90-degree
0.000 0.005 0.010 0.015 0.020
True Strain [-]
0 20 40 60 80 100
True Stress [MPa] 0-degree
45-degree 90-degree
Figure 4.6: Numerical (dashed line) and experimental comparison of engineering stress-strain curves (left) and true stress-strain curves (right).